On limit behavior of semigroup actions on noncompact spaces
HTML articles powered by AMS MathViewer
- by Josiney A. Souza
- Proc. Amer. Math. Soc. 140 (2012), 3959-3972
- DOI: https://doi.org/10.1090/S0002-9939-2012-11248-1
- Published electronically: March 23, 2012
- PDF | Request permission
Abstract:
This paper is produced in response to the questioning of Morse decomposition for semigroup actions on noncompact spaces. We show how the limit behavior can be studied in arbitrary topological spaces by using powerful tools such as the Stone-Čech compactification and shadowing semigroups. We extend Conley’s characterization of chain recurrence in terms of attractors from the setting of flows on compact metric spaces to the setting of semigroup actions on any topological space.References
- Carlos José Braga Barros and Luiz A. B. San Martin, Chain control sets for semigroup actions, Mat. Apl. Comput. 15 (1996), no. 3, 257–276 (English, with English and Portuguese summaries). MR 1429552
- Carlos J. Braga Barros and Luiz A. B. San Martin, Chain transitive sets for flows on flag bundles, Forum Math. 19 (2007), no. 1, 19–60. MR 2296065, DOI 10.1515/FORUM.2007.002
- Carlos J. Braga Barros and Josiney A. Souza, Attractors and chain recurrence for semigroup actions, J. Dynam. Differential Equations 22 (2010), no. 4, 723–740. MR 2734477, DOI 10.1007/s10884-010-9164-3
- Carlos J. Braga Barros and Josiney A. Souza, Finest Morse decompositions for semigroup actions on fiber bundles, J. Dynam. Differential Equations 22 (2010), no. 4, 741–760. MR 2734478, DOI 10.1007/s10884-010-9165-2
- Braga Barros, C. J., Souza, J. A. and Reis, R. A. (2012). Dynamic Morse decompositions for semigroups of homeomorphisms and control systems. J. Dyn. Control Syst. 18, 1-19.
- Sung Kyu Choi, Chin-Ku Chu, and Jong Suh Park, Chain recurrent sets for flows on non-compact spaces, J. Dynam. Differential Equations 14 (2002), no. 3, 597–611. MR 1917652, DOI 10.1023/A:1016339216210
- Fritz Colonius and Wolfgang Kliemann, The dynamics of control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000. With an appendix by Lars Grüne. MR 1752730, DOI 10.1007/978-1-4612-1350-5
- Mike Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 709–729. MR 1145617, DOI 10.1017/S014338570000643X
- Mike Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1139–1148. MR 1098401, DOI 10.1090/S0002-9939-1992-1098401-X
- Mike Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations 7 (1995), no. 3, 437–456. MR 1348735, DOI 10.1007/BF02219371
- Mike Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc. 126 (1998), no. 1, 245–256. MR 1458880, DOI 10.1090/S0002-9939-98-04500-6
- Mauro Patrão, Morse decomposition of semiflows on topological spaces, J. Dynam. Differential Equations 19 (2007), no. 1, 181–198. MR 2279951, DOI 10.1007/s10884-006-9033-2
- Mauro Patrão and Luiz A. B. San Martin, Semiflows on topological spaces: chain transitivity and semigroups, J. Dynam. Differential Equations 19 (2007), no. 1, 155–180. MR 2279950, DOI 10.1007/s10884-006-9032-3
- Mauro Patrão and Luiz A. B. San Martin, Morse decomposition of semiflows on fiber bundles, Discrete Contin. Dyn. Syst. 17 (2007), no. 3, 561–587. MR 2276428, DOI 10.3934/dcds.2007.17.561
- Verdi, M. A., Rocio, O. G. and San Martin, L. A. B. (2009). Semigroup Actions on Adjoint Orbits. To appear.
- Stephen Willard, General topology, Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581]. MR 2048350
Bibliographic Information
- Josiney A. Souza
- Affiliation: Departamento de Matemática, Universidade Estadual de Maringá, Maringá 87020-900, Brasil
- Email: jasouza3@uem.br
- Received by editor(s): February 10, 2011
- Received by editor(s) in revised form: May 12, 2011
- Published electronically: March 23, 2012
- Communicated by: Yingfei Yi
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3959-3972
- MSC (2010): Primary 37B35, 37B25
- DOI: https://doi.org/10.1090/S0002-9939-2012-11248-1
- MathSciNet review: 2944735